Sir Chris Hoy leads the GB Cycling Team during the official opening of the Velodrome (Photograph by David Poultney)

It's the very last medal day for track cycling! If you've been watching, you'll have noticed just how steeply the elegant wooden cycling track is banked in the turns. Echoing this shape is the sweeping curved roof of the Velodrome (which hit the headlines in its own right during the torrential rain this weekend) topping the beautiful cedar clad exterior. To find out why and how the track and its buildings came to have the shapes they do, read the Plus articles Leaning into 2012 and How the velodrome found its form.

Yesterday was a great day for badminton with gold medals being awarded in the men's singles and doubles. This got us thinking about shuttlecocks. They are not like balls at all and this means that they don't behave like balls either. John D. Barrow, mathematician, cosmologist and prolific popular science writer, explains.

Shuttlecocks used for badminton are not like other projectiles found in sports. They are extremely asymmetrical, with a conical skirt about 6cm long and 6cm across that is attached to a cork of higher density at the narrow end of the cone. However, when the shuttlecock is hit it will quickly flip over so that the cork is leading because its pressure centre if different to its centre of mass. This will ensure that it is always approaching your opponent the right way round for her to hit it back.

The effect of hitting the shuttlecock is strange. When you hit a tennis ball or a cricket ball with a racquet or bat it goes further the harder you hit it. But no matter how hard you hit the shuttlecock it won't go much further than about 6 or 7 metres.

The trajectory of a shuttlecock, moving from left to right: it falls steeper than it rises.

The motion of the shuttlecock obeys Newton's laws of motion. Its acceleration is governed by the downward force of gravity and a drag force from the air that is proportional to the square of the shuttlecock's speed through the air. When it is first struck the shuttlecock is moving at its fastest and the drag force is therefore much bigger than gravity. So it moves upwards in a straight line, at an angle determined by the direction of impact from the racquet, gradually being decelerated by gravity. Eventually it is going so slowly that the force of gravity is comparable to the air drag force and the trajectory reaches a maximum height and curves downwards towards the ground. Gravity is now speeding it up and it quickly reaches a speed, termed the terminal speed, where the opposing forces of gravity (downwards) and air drag (upwards) become equal. There is now no net force on the shuttlecock and it moves downwards at this constant terminal speed, without experiencing any acceleration or deceleration. The overall trajectory doesn't look like a parabola: the shuttlecock falls steeper than it rises.

The terminal speed does not depend on the initial launch speed of the shuttlecock. It is determined by the strength of gravity, air density, the size and mass of the shuttlecock, and its smoothness. As a result it is these unchanging properties that fix the distance that the shuttlecock will reach when struck hard. Hitting it even harder can't make it go any further than these properties dictate.

Image: Jmex60.

Usain Bolt is determined to become a legend this weekend, by running the 100m in 9.4 seconds. But what does mathematics have to say about this quest? What is the ultimate limit which no runner can possibly surpass? How can Bolt improve his record significantly without improving his speed? What about the effects of wind assistance, timing accuracy, and altitude on sprint times? Find out about all this and more in the Plus article No limits for Usain and in this video of a lecture given by John D. Barrow and featured on our sister site Maths and sport: Countdown to the Games.

The weekend will also be a great one for Olympic tennis so you should rehearse the scoring rules:

You win a game if you score 4 points before your opponent scores 3 points. Or, if you both score 3 points at some stage you win if you manage to score 2 points in a row after the 3-all stage before your opponent does.

That's quite a mouthful and it turns out that the maths behind tennis can get tricky too. What is the secret to the perfect serve? Can you figure out the probability of winning a game when the probability of winning a point is 0.6? What's the chance of a tennis match taking on epic proportions like the 11-hour battle between John Isner and Nicolas Mahut at Wimbledon in 2011? Why does tennis use this comparatively complicated scoring system? And can you improve your chance of winning by choosing the right racket? Here are some answers to these questions, from Plus and from our sister site Maths and sport: Countdown to the Games:

Spinning the perfect serve — A new mathematical analysis of how to hit a winning server shows that spin is the thing. Perhaps there's still time for Murray's coach to include some maths in his preparations for the match today...

Any win for tennis? — Work out the probability of winning a game for a fixed probability of winning a point. This challenging activity is designed to be accessible to students of A-level maths and anyone else who likes puzzling over probabilities.

Anyone for tennis (and tennis and tennis ...)? — What's the chance of a tennis match taking on epic proportions like the 11 hour battle between John Isner and Nicolas Mahut at Wimbledon in 2011?

Final score — Why are there so many different scoring systems in operation in sport? This video of a lecture given by John D. Barrow looks at how structuring matches into a series of sets affects the relative roles of luck and skill in determining the winner of the contest. It also looks at issues surrounding scoring in table tennis and decathlon.

Making a racket: the science of tennis — While the players get most of the limelight, engineers, too, are working hard to produce the cutting-edge tennis rackets that guarantee record performances. Over recent decades new materials have made tennis rackets ever bigger, lighter and more powerful. So what kind of science goes into designing new rackets?

Yesterday rowers Helen Glover and Heather Stanning won Great Britain's first gold medal. And the men's eight claimed a bronze with world champions Germany taking gold. Could the men's team have done better if they'd arranged their oars differently? Usually you expect to find rowers positioned in a symmetrical fashion, alternately right-left, right-left as you go from one end of the boat to the other. However, the regularity of the rower's positions hides a significant asymmetry that affects the way the boat will move through the water. Find out more in Rowing has its moments.

And if you prefer watching to reading, our sister site Maths and sport: Countdown to the Games features a video of a lecture given by John D. Barrow, which explores how to rig a rowing eight, whether a cox helps or hinders a racing boat, how the speed of a kayak or a canoe depends on the number of paddlers and what happens if you fall in.